Nonlinear evolution equations and dynamical systems

A series of international workshops

The NEEDS Series of international meetings began in 1980. The
original motivation was twofold: on one hand the ambition to
contribute to the major development in theoretical and mathematical
physics, in applied mathematics, in pure mathematics and in several
applicative domains, associated with the discovery in the 70s of many
integrable systems (classical and quantum) and of various techniques
for their study and solution (the Soliton revolution); on the other,
the intention to foster exchanges among scientists from the “West”
and the “East” (including, in the latter case, the former
Soviet Union). The second goal appeared particularly crucial, since
the contributions to this research field by scientists in the former
Soviet Union were absolutely outstanding, yet many of those who had
provided and were producing such contributions had enormous
difficulties in making contact with their colleagues living elsewhere
who were working in the same area - and, of course, viceversa.

This series of meetings was quite successful in fulfilling both
goals: the participation of scientists from the former Soviet Union,
East Europe, China – together with that of scientists from
Western Europe, the USA, Japan, Canada, Australia,... - has been
substantial, already in the early years when travelling to the West
was very difficult for many scientists, especially from the former
Soviet Union; in several cases, the NEEDS meetings have provided the
first opportunity for outstanding researchers from the Eastern
countries to travel abroad, beyond the Iron Curtain. Through the 80s
and the 90s, when the field of nonlinear integrable equations has
continued to witness a major development - possibly the most
interesting and vital development in theoretical and mathematical
physics and applied mathematics of the second half of the 20th
century - these international meetings have significantly fostered
the blooming of this research area.

The interdisciplinary character of the theory of integrable equations
can be traced back to its earliest days. Historically, there exist
different and independent beginnings in fields that are as far apart
as hydrodynamics (Russell's great wave of translation, 1834; Korteweg
- de Vries equation, 1895), differential geometry (pseudo-spherical
surfaces, 1839), mechanics (Kowalevski top, 1889) solid state physics
(Frenkel-Kontorova model, 1938) and numerics (Fermi-Pasta-Ulam
experiment, 1955). The breakthrough occurred in 1967, thanks to the
discovery of the appropriate technique (a kind of nonlinear
generalization of the Fourier transform) to treat certain classes
of evolution partial differential equations and thereby to fully
understand the role of a new nonlinear coherent structure, the
soliton. From then on the discipline combined all formerly
independent roots and became the interdisciplinary subject that it is
today. The NEEDS meetings, from the beginning of the 1980s, played a
crucial role in this process: they provided the global opportunity of
discussion and interaction for a scientific community who was
developing a new field of research. A turning point in the theory of
integrable equations was the exciting discovery, in the late sixties,
that several partial differential equations possess a variety of
nontrivial exact solutions. These include not only solitary waves,
known since the nineteenth century, but also solutions involving an
arbitrary number of solitary waves of varying speeds and amplitudes
undergoing collective collisions. The solitary waves are now called
solitons to emphasize their particle-like character, since they leave
the interaction region of space-time with the same shape they had
upon entry. These phenomena are not just mathematical curiosities,
bearing no relation to reality; quite on the contrary, the most
significant soliton systems arise in the context of outstanding
problems in applied science.

After these discoveries a large amount of research was devoted to the
study of nonlinear processes; in this context many important
nonlinear equations turned out to have a universal character,
thereby explaining the remarkable fact that they were both integrable
and widely applicable. The NEEDS meetings provided
not only the opportunity for the world’s leading scientists and
young researchers to meet in an informal setting for discussions at
the highest level on recent developments in the field, they acted
moreover as a catalyst for creating new synergistic contacts
throughout Europe and the rest of the world. Through the 80s
and the 90s one of the main lines of research was devoted to discover
new integrable systems and to find appropriate methods to solve them
(Inverse Scattering Transform, Direct Linearization Method,
Dressing Method, Hirota Bilinear Technique,...). Soon afterwards it
became important to classify known equations and to develop
techniques to recognise whether a given nonlinear system is
integrable or not (symmetry approach, tests of integrability such as
the Painleve’- Kowalevski test, necessary conditions of
integrability,...). The mathematical structures underlying
integrability were investigated using analytical, algebraic and
geometrical approaches. In this context, new mathematical objects
such as recursion operators, bi-Hamiltonian structures, reduction
groups, master symmetries were introduced. These outstanding
scientific results have been discussed at the NEEDS workshops and
some of them have indeed been obtained during these workshops
themselves.

Despite the major progress, much remains to be done. Nonlinearity
constitutes a central theme of current research in theoretical and
mathematical physics, as well as in applied and pure mathematics; and
it plays a crucial role in several applied fields.

Besides their main scientific goal, the NEEDS meetings face nowadays
a new organizational challenge: to increase the participation of
outstanding researchers from developing countries.